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Topics in Normal Bases of Finite Fields N. A. Carella

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Topics in Normal Bases of Finite Fields N. A. Carella Topics in Normal Bases of Finite Fields N. A. Carella Copyright 2001. All rights reserved. Table of Contents Chapter 1 Bases of Finite Fields 1.1 Introduction……………………………………………………………………………….2 1.2 Definitions and Elementary Concepts………………………………………………….....3 1.3 The Discriminants of Bases……………………………………………………………….7 1.4 Distribution of Bases…………………………………………………………………….10 1.5 Dual Bases……………………………………………………………………………….11 1.6 Distribution of Dual Bases…………….………………………………………………...16 1.7 Polynomials Bases……………………………………………………………………….17 Chapter 2 Structured Matrices 2.1 Basic Concepts…………………………………………………………………………..22 2.2 Circulant Matrices……….………………………………………………………………24 2.3 Triangular Matrices……………………………………………………………………...31 2.4 Hadamard Matrices……………………………………………………………………33 2.5 Multiplication Tables…………………………………………………………………….35 Chapter 3 Normal Bases 3.1 Basic Concepts………………………………………….……………………………….40 3.2 Existence of Normal Bases………………………………..……………………………..41 3.3 Iterative Construction of Normal Bases………………….……………………………...42 3.4 Additive Order and Decomposition Theorem…………….……………………………..44 3.5 Normal Tests……………………………………………….……………………………46 3.6 Polynomial Representations………………………………….………………………….48 3.7 Dual Normal Bases……………………………………………..………………………..51 3.8. Distribution Of Normal Bases…………………………………..………………………53 3.9 Distribution Of Self-Dual Normal Bases…………………………………………...…54 3.10. Formulae For Normal Elements/Polynomials………………….…………………..…57 3.11 Completely Normal Bases and Testing Methods……………………………………59 3.12 Infinite Sequences of Normal Elements/Polynomials………………………………...62 3.13 Characteristic Functions……………………………………………..…………….…64 3.14 Primitive Normal Bases……………………………………………..…………….…67 3.15 Applications Of Fractional Linear Transformations To Normal Bases…………….…68 Chapter 4 General Periods 4.1 Concept of Periods………………………………………………………………………72 4.2 Cyclotomic Periods…………………………………………………………………...…74 4.3 Cyclotomic Numbers…………………………………………………………………….76 4.4 Linear and Algebraic Properties of the Integers (i, j)……………………………………79 4.5 Characterization of the Periods………………………………………..………………80 4.6 Cyclotomic Numbers of Short Type……………………………………………………..81 4.7 Extension to the Ring Zr……………………………………..………..………………86 4.8 Extension of the Finite Field Fp………………………………………..……………...86 4.9 Exponential Sum Properties………………………………………….……………….87 Chapter 5 Periods Polynomials 5.1 Definition of Period Polynomials………………………………………………………92 5.2 Discriminant and Factorization of Period Polynomials…………………………………94 5.3 Period Polynomials of Low Degrees…………………………………………………..95 5.4 Period Polynomials of High Degrees…………………………………………………..101 5.5 Coefficients Calculations Via Power Sums Method…………………………………107 5.6 Sequences of Period Polynomials……………………………………………………110 Chapter 6 Periods Normal Bases 6.1 Definitions and Existence …….……………………………………………………114 6.2 Existence of Period Normal Bases……………………… …………………………115 6.3 Polynomial Representations of Periods…………………………….………………120 6.4 Dual Period Normal Bases………………………………………………….………122 Chapter 7 Period Normal Bases For Extensions Of Low Degrees 7.1 Quadratic Extensions……………………………………………………………….128 7.2 Dual Period Normal Bases of Quadratic Extensions ………...…………………….129 7.3 Multiplications in Quadratic Extensions……………………………………………131 7.4 Criteria For Cubic Nonresidues…………………………………………………….135 7.5 Period Normal Bases of Cubic Extensions…………………………………………136 7.6 Dual Period Normal Bases of Cubic Extensions……………………...……………138 7.7 Multiplications in Cubic Extensions………………………………………………..138 Chapter 8 Asymptotic Proofs 8.1 Primitive Polynomial with Prescribe Coefficients…………………………………..143 8.2 Primitive Normal Polynomial with Prescribe Coefficients ………………….……..153 References…………………………………………………………………………….165 Chapter 1 Bases of Finite Fields Bases of Finite Fields 1.1 Introduction This chapter introduces various fundamental ideas and terminologies essential for the understanding of vector representations of finite fields. The study of bases of vector space representations of finite fields and the corresponding computational algorithms is an extensive and important subject. There are various methods of representing finite fields. The most common are vector spaces, cyclic representations, polynomial quotient rings, quotients of number fields, matrix representations, and binary representations respectively. These are listed here in order. ≅ ⋅⋅⋅ ∈ (1) Fqn { x = x0α0 + x1α1 + + xn−1αn−1 : xi Fq }, where {α0, α1, ..., αn−1} is a basis. ≅ ∪ (2) Fqn < ξ > {0}, where ξ is a generator of the multiplicative group of Fqn . ≅ (3) Fqn Fq[x]/(f(x)), where f(x) is an irreducible polynomial of degree n. ≅ (4) Fqn OK/(ℐ), where ℐ is a maximal ideal and OK is the ring of integers in a numbers field K. These representations are widely used in algebraic number theory. ≅ (5) Fqn { Subset of Nonsingular Matrices }. ≅ − φ → n (6) Fqn { l adic Vectors }, the vectors are defined by a function : Fqn Fl . Two instances are the 2−adic representation (binary): qn −1 (qn −1) / 2 qn −1 (qn −1) / 2 φ(x) = [(x + x +α n−1 ) / 2, ..., (x + x +α 0 ) / 2] , and the 3−adic representation: (qn −1) / 2 (qn −1) / 2 (qn −1) / 2 φ(x) = [(x +α n−1 ) , ..., (x +α1 ) , (x +α 0 ) ] α α α ∈ n where 0, 1, …, n−1 Fq are fixed. The fastest methods for addition and subtraction are implemented with vector space representations. And the fastest method for multiplications, divisions, discrete exponentiations, and certain root extractions are implemented with cyclic representations. In the other hand, the fastest algorithms for computing discrete logarithms in finite fields are implemented in polynomial quotient rings and quotient of number fields, see [1, Adleman and DeMarrais], [1, ElGammal] etc. Matrix representations have applications in the construction of hash functions, Copyright 2001. - 2 - Bases of Finite Fields pseudo numbers generators, and others, see [3, Gieselmann]. Normal bases, (which are vector space representations), and Binary representations are useful in polynomial factorizations, see [1, Nieterreiter], [1, Camion], and [1, Ganz]. Since additions and subtractions are highly efficient operations with respect to most bases, the main focus is on multiplication and multiplicative inverse algorithms with respect to the various bases of the finite fields Fqn over Fq. 1.2 Definitions and Elementary Concepts Several methods for identifying the bases of the vector space Fqn over Fq will be considered in this section. The notion of basis of a vector space has already appeared in this text. The concept of basis and the related idea of linear independence, (also algebraic independence), are recurrent themes throughout mathematics. ⊂ Definition 1.1. A subset of elements {α0, α1, ..., αn−1} Fqn is said to be a basis of the vector ∈ space Fqn over Fq if and only if every element α Fqn can be uniquely written as a linear combination α = a0α0 + a1α1 + ⋅⋅⋅ + an−1αn−1, where ai ∈ Fq. A redundant basis is a basis such that every element has a representation as linear combination but not necessarily unique. Some algorithms based on redundant bases are more efficient than those based on nonredundant bases. A redundant normal basis {α0, α1, ..., αn−1, 1} of Fqn over Fq is employed in [2, Gao, et al] to improve the exponentiation algorithm. A redundant normal basis permits multiple representations of the elements, e.g., 0 = α0 + α1, + ⋅⋅⋅ + α−n1 + 1 among others if q > 2. Redundant bases are also used to represent integers in fast exponentiation algorithms, and real/complex numbers in numerical algorithms which implement carry free arithmetic operations. Definition 1.2. Let {α0, α1, ..., αn−1} be a subset of Fqn . The regular matrix representation of i qi the set {α0, α1, ..., αn−1} is defined by the n×n matrix A = ( σ (α j) ) = ( α j ) . The matrix Copyright 2001. - 3 - Bases of Finite Fields α0 α1 α2 . αn−1 q q q q α0 α1 α2 . αn−1 A = q2 q2 q2 . q2 α0 α1 α2 αn−1 . qn−1 qn−1 qn−1 . qn−1 α0 α1 α2 αn−1 occurs very frequently in the analysis of bases and matrix representations of linear functionals in finite fields. Definition 1.3. A pair of bases {α0, α1, ..., αn−1} and {β0, β1, ..., βn−1} are equivalent if each βi = cαi for some constant c ∈ Fq. The equivalence class of each basis {α0, α1, ..., αn−1} can be viewed as a point in (n−1)- n−1 dimensional projective space P (Fqn ). Lemma 1.4. A subset {α0, α1, ..., αn−1} of elements of Fqn is a basis of the vector space Fqn over Fq if and only if the n×n regular matrix representation A associated to {α0, α1, ..., αn−1} is nonsingular. ∈ Proof: Suppose that {α0, α1, ..., αn−1} is a basis, and let β Fqn . Now consider the system of equations n−1 n−1 n−1 q q qn−1 qn−1 β = ∑biαi , β = ∑biαi , ..., β = ∑biαi . i=0 i=0 i=0 Since the subset of elements {α0, α1, ..., αn−1} is a basis, the system of equations, rewritten as a vector equation n−1 (β, β q , ..., β q )= Ab . has a unique solution b = (b0,b1,...,bn−1). This implies that the matrix A is nonsingular. Conversely, if the matrix A is nonsingular, then the above system of equations has a unique ∈ solution. This in turn implies that each β Fqn has a unique representation as a linear combination β = b0α0 + b1α1 + ⋅⋅⋅ + bn−1αn−1, bi ∈ Fq, so {α0, α1, ..., αn−1} is a basis. Lemma 1.5. (Basis lifting lemma) A basis {α0, α1, ..., αn−1} of Fqn over Fq is also a basis of Fqnk over Fqk for all integers k such that gcd(k, n) = 1. Copyright 2001. - 4 - Bases of Finite Fields ∈ Proof: Let a0, a1, ..., an−1 Fqk and consider the system of equations a0 α0 + a1α1 + a2 α2 + + an−1αn−1 = 0 qk qk qk qk a0 α0 + a1α1 + a2 α2 + + an−1αn−1= 0 q2k q2k q2k q2k a0 α0 + a1α1 + a2 α2 + + an−1αn−1= 0 . q(n−1 )k q(n−1 )k q(n−1 )k q(n−1 )k a0 α0 + a1α1 + a2 α2 + + an−1αn−1 = 0 Since gcd(k, n) = 1, the map i → ik is a permutation of {0, 1, 2, ..., n−1} and the matrices qi qik A = ( j ) and Ak = ( α j ) are just rows permutations of each other. Moreover, because the regular matrix representation A attached to this basis is nonsingular, it follows that the system of equations has only a trivial solution a = (a0,a1,...,an−1) = (0,0,...,0). This proves the linear independence of {α0, α1, ..., αn−1} over Fqk .
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